Freedom, Hierarchies and Saturated Regression (WIP)

import pandas as pd
import numpy as np
import bambi as bmb
import pymc as pm
import arviz as az
import matplotlib.pyplot as plt
from matplotlib import transforms
from itertools import product
import pyfixest as pf
import statsmodels.formula.api as smf
from patsy import dmatrices, dmatrix
from pyfixest.did.estimation import did2s, lpdid
import warnings
warnings.simplefilter(action='ignore', category=FutureWarning)
pd.options.mode.chained_assignment = None 

np.random.seed(100)

Two-way fixed effects estimators in Difference in Differences designs can be seen as a species of Mundlak regression. We unpack what this means and what it implies for model development in causal inference.

Summary

WORK IN PROGRESS

Inference, causal and otherwise, is biased in the presence of implicit group level confounding. Regression modelling can account for these risks but we have to appropriately condition our inference steps on the right group level effects. It is not always obvious how to incorporate these factors in our modelling. Fixed Effects and (Hierarchical) Random Effects are variously use to to express variation in the outcome due to group level effects. Additionally we’ll see how Mundlak devices can be used to express variants of these models. Questions of correct functional form become more complicated again when we have to account for variation across multiple groups interactions. We explore these questions in assessing class/school/pupil dynamics before applying these considerations to causal inference more directly.

Two Way Fixed Effects (TWFE) regression models are often used in Differences-in-Differences causal inference designs to estimate treatment effects while accounting for variation due to group effects in two dimensions. We will explore the relationship between TWFE and Mundlak regressions in DiD and Event Study designs, showing how sound inferential strategies rely on saturated models with richly parameterised specifications. In particular we’ll see how TWFE breaks down in cases of staggered treatment regimes, and how to fix it.

Throughout we’ll show how the different model architectures promote different interepretations of the group confounding and conclude by teasing out some of these implications for model choice.

Architectures and Free Parameters

“There are no rules of architecture for a castle in the clouds.” - G.K. Chesterton

Free choice is not whimsy or wanton action. Choice is free when taken within the confines of tolerable bounds. In statistical modelling we seek to discover what is plausibly realised within such bounds. We learn from the past to discern the shape of the processes which determine the future. Different models yield different insights into those structures which govern our trajectories. Which parameters are “free”, and which are decisively characterised? Which realities are contestable and which non-negotiable? In this post we will dive into the way in which different inferential methodologies yield different insights into how group-level confounding constrains the realisations of our parameter space and shapes the architecture of causal inference.

Difference in Differences estimation is core a topic in contemporary social science and causal inference. It is a compelling methodology for licensing inferential statements about policy effects, but it is not a panacea. The method makes substantive assumptions about the treatment regime and the trends in the process before the treatment is introduced, but it also requires that the data is rich enough to mandate the intended interpretation of a heavily parameterised estimation model. The two-way fixed effects models attempt to tightly estimate the treatment effect by controlling for all sources of variation in our treatment due to group level factors in the DiD designs. These controls isolate the variation due to actual treatment by pinning down the effects of interactions between treatment-units and time. We will see how these models and their intended interpretation come apart quickly in the face of staggered treatment roll-outs. We will also show how to fix these issues.

Structure of the Presentation

In this post we will structure the discussion as follows:

• Estimation Strategies and Group Effects
  • We go through a paired down parameter recovery examples of group-confounding and a range of estimation strategies aiming to account for group effect bias.
• Nested Groups, Hierarchies and Fixed Effects
  • We work through a real example of three-level group nesting; pupils in classes, in schools. Where we need to make decisions about the estimation strategy.
• Two Way Fixed Effects and Temporal Confounding
  • We branch off into examining an analogous case of confouning addressed in DiD designs. We show how TWFE works and how to specify these regression designs using Fixed Effects and Mundlak devices.
• Issues with TWFE and Richly Parameterised Linear Models
  • We examine the problems arising for the TWFE estimator in the case of staggered roll-outs. We also see how to address the issue with Mundlak devices, and reflect on what we’ve seen.

Estimation Strategies and Group Effects

First consider an estimation example due to Richard McElreath’s lecture series where we examine the various parameter recovery options available in the case of group level confounding. Define a data generating process determined by group level effects:

def inv_logit(p):
    return np.exp(p) / (1 + np.exp(p))

N_groups = 30
N_id = 3000
a0 = -2
bXY = 1
bZY = -0.5
g = np.random.choice(range(N_groups), size=N_id, replace=True)
Ug = np.random.normal(1.5, size=N_groups)

X = np.random.normal(Ug[g], size=N_id)
Z = np.random.normal(size=N_groups)

s = a0 + bXY*X + Ug[g] + bZY*Z[g]
p = inv_logit(s)
Y = np.random.binomial(n=1, p=p)


sim_df = pd.DataFrame({'Y': Y, 'p': p, 's': s, 'g': g, 'X': X, 'Z': Z[g]})
sim_df.head()

	Y	p	s	g	X	Z
0	1	0.943243	2.810542	8	1.384072	-1.818302
1	0	0.593872	0.379996	24	1.047128	-0.242817
2	1	0.874945	1.945406	3	2.340418	-0.650122
3	0	0.317089	-0.767181	7	0.550504	0.795680
4	1	0.699030	0.842684	23	1.183011	-0.111692

This data generating process is characterised by the bernoulli outcome with group level confounds. If we model this relationship the confounding effects will bias naive parameter estimates on the covariates \(X\), \(Z\). We will see different results as we explore different ways of parameterising the relationships, and the trade-offs for model specifications will become clearer.

Warning

There is a huge degree of confusion over the meaing of the terms “Fixed Effects” and “Random Effects”. Within this blog post we will mean the population level parameters. \[\beta X\]

for an individual variable \(X\) when we refer to fixed effects. Corresspondingly we will refer to group-level parameters \(\beta_{g}\)

\[\Big(\underbrace{\beta}_{pop} + \underbrace{\beta_{g}}_{group}\Big)X\]

which are incorporated into our model equation modifying population level parameters as random effects. We will generally use Wilkinson notation to specify these choices where random effects for modifying a population are denoted with a conditional bar over the variable ( X | group) and fixed effects are specified by just including the variable in the equation i.e. y ~ X + (Z | group) where X has a fixed effect parameterisation and Z a random effects parameterisation. We can also create indicator variables for group membership using this syntax with y ~ C(group) + X + Z where under the hood we pivot the group category into a zero-one variables indicating group membership. This parameterisation means each indicator variable (one for each level of the grouping variable) will receive a fixed effects population parameter.

Naive Model

The formula syntax for Bambi also enables us to specify generalised linear models with bernoulli outcomes.

naive_model = bmb.Model(f"Y['>.5'] ~ X + Z ", sim_df, 
family="bernoulli")
naive_idata = naive_model.fit( inference_method="nuts_numpyro",
    idata_kwargs={"log_likelihood": True},)

az.summary(naive_idata, var_names=['Intercept', 'X', 'Z'])

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
Intercept	-0.992	0.076	-1.142	-0.859	0.001	0.001	5492.0	3094.0	1.0
X	1.309	0.053	1.217	1.413	0.001	0.001	3294.0	3007.0	1.0
Z	-0.613	0.048	-0.707	-0.526	0.001	0.001	3137.0	2897.0	1.0

Here we see that all three parameter estimates are biased away from their true values. Let’s try a simple fixed effects approach that adds indicator variables for all but one of the group levels.

Fixed Effects Model

The additional syntax for creating the group level indicator variables is specified here.

fixed_effects_model = bmb.Model(f"Y['>.5'] ~ C(g) + X + Z", sim_df, 
family="bernoulli")
fixed_effects_idata = fixed_effects_model.fit( inference_method="nuts_numpyro",
    idata_kwargs={"log_likelihood": True},)

az.summary(fixed_effects_idata, var_names=['Intercept', 'X', 'Z'])

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
Intercept	-0.185	1.571	-3.306	2.644	0.082	0.058	369.0	479.0	1.0
X	0.977	0.062	0.865	1.098	0.001	0.001	2592.0	2370.0	1.0
Z	-0.187	1.469	-2.883	2.678	0.076	0.054	373.0	521.0	1.0

Now we see that the coefficient on the \(X\) variable seem correct, but the coefficient on \(Z\) is wildly wrong. Indeed the uncertainty interval on the \(Z\) coefficient is huge. The fixed effect model was unable to learn anything about the correct parameter. Whereas the naive model seems to learn the correct \(Z\) parameter but over estimates the \(X\) coefficient. These are the kinds of trade-offs we need to be wary of as we account for the complexities of extensive group interactions in our model’s functional form.

Code

fig, axs = plt.subplots(2, 1, figsize=(10, 9))
axs = axs.flatten()

az.plot_posterior(naive_idata, var_names=['X'], ax=axs[0], 
point_estimate=None,  hdi_prob='hide', color='red', label='Naive Model')
axs[0].axvline(1, color='k')

az.plot_posterior(fixed_effects_idata , var_names=['X'], ax=axs[0], point_estimate=None, hdi_prob='hide', label='Fixed Effect Models')
axs[0].set_title("Naive/Fixed Model X Coefficient")

az.plot_posterior(naive_idata, var_names=['Z'], ax=axs[1], point_estimate=None,  hdi_prob='hide', color='red', ref_val_color='black')


az.plot_posterior(fixed_effects_idata , var_names=['Z'], ax=axs[1], point_estimate=None, hdi_prob='hide')

axs[1].set_title("Naive/Fixed Effect Model Z Coefficient")
axs[1].axvline(-0.5, color='k');

We now want to try another approach to handle to the group confounding that involves a hierarchical approach to add group level effects to the intercept term.

Multilevel Model

The syntax for the random effects model is similar.

multilevel_model = bmb.Model(f"Y['>.5'] ~ (1 | g) + X + Z", sim_df, 
family="bernoulli")
multilevel_model_idata = multilevel_model.fit( inference_method="nuts_numpyro",
    idata_kwargs={"log_likelihood": True},)

This method posits a view that there is a shared underlying process generating each instance of group-behaviour i.e. the realised values of the outcome within each group. In the random effects multilevel model we replace the indivdual fixed effects indicator columns with additional parameters modifying the intercept term. This is the “hierarchy” or multi-level aspect of the specification.

az.summary(multilevel_model_idata, var_names=['X', 'Z', 'Intercept'])

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
X	1.023	0.060	0.911	1.137	0.001	0.001	4182.0	3164.0	1.00
Z	-0.592	0.163	-0.894	-0.287	0.005	0.004	915.0	1543.0	1.00
Intercept	-0.570	0.196	-0.937	-0.194	0.007	0.005	761.0	1334.0	1.01

Next we’ll apply the Mundlak device method which adds the group mean back to each observation as a covariate.

Mundlak Model

For this technique we supply group aggregate mean values per group as an additional column. This somewhat absolves of the requirement to include an extra multiplicity of parameters. It’s more akin to feature creation than model specification, but it serves the same purpose - it accounts for group level variation and provides a mechanism for the model to learn the appropriate weights to accord each group in the final calculation.

sim_df['group_mean'] = sim_df.groupby('g')['X'].transform(np.mean)

sim_df['group_mean_Z'] = sim_df.groupby('g')['Z'].transform(np.mean)

mundlak_model = bmb.Model(f"Y['>.5'] ~ (1 | g) + X + Z + group_mean", sim_df, 
family="bernoulli")
mundlak_idata = mundlak_model.fit( inference_method="nuts_numpyro",
    idata_kwargs={"log_likelihood": True},)

az.summary(mundlak_idata, var_names=['Intercept', 'X', 'Z'])

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
Intercept	-1.903	0.138	-2.165	-1.657	0.002	0.002	3375.0	3148.0	1.0
X	0.952	0.059	0.837	1.059	0.001	0.001	4496.0	2974.0	1.0
Z	-0.446	0.063	-0.567	-0.327	0.001	0.001	2728.0	2405.0	1.0

We can now plot all the parameter recovery models together and we’ll see that there are some trade-offs between the fixed effects and random effects varieties of the modelling.

Plotting the Comparisons

The parameter recovery exercise shows striking differences across each of the models

Code

fig, axs = plt.subplots(3, 1, figsize=(10, 11))
axs = axs.flatten()

az.plot_posterior(naive_idata, var_names=['X'], ax=axs[0], 
point_estimate=None, color='red', label='Naive', hdi_prob='hide')
axs[0].axvline(1, color='k', linestyle='--', label='True value')

az.plot_posterior(fixed_effects_idata , var_names=['X'], ax=axs[0], point_estimate=None, hdi_prob='hide', label='Fixed')

az.plot_posterior(multilevel_model_idata, var_names=['X'], ax=axs[0], point_estimate=None, hdi_prob='hide', color='green', label='Hierarchical')

az.plot_posterior(mundlak_idata, var_names=['X'], ax=axs[0], point_estimate=None, hdi_prob='hide', color='purple', label='Mundlak')


axs[0].set_title("X Coefficient")

az.plot_posterior(naive_idata, var_names=['Z'], ax=axs[1], point_estimate=None, hdi_prob='hide',  color='red', ref_val_color='black')


az.plot_posterior(fixed_effects_idata , var_names=['Z'], ax=axs[1], point_estimate=None, hdi_prob='hide')

az.plot_posterior(multilevel_model_idata, var_names=['Z'], ax=axs[1], point_estimate=None, hdi_prob='hide', color='green')

az.plot_posterior(mundlak_idata, var_names=['Z'], ax=axs[1], point_estimate=None, hdi_prob='hide', color='purple')

axs[1].set_title("Z Coefficient")
axs[1].axvline(-0.5, color='k', linestyle='--');

az.plot_posterior(naive_idata, var_names=['Intercept'], ax=axs[2], point_estimate=None,  color='red', ref_val_color='black', hdi_prob='hide')


az.plot_posterior(fixed_effects_idata , var_names=['Intercept'], ax=axs[2], point_estimate=None, hdi_prob='hide')

az.plot_posterior(multilevel_model_idata, var_names=['Intercept'], ax=axs[2], point_estimate=None, hdi_prob='hide', color='green')

az.plot_posterior(mundlak_idata, var_names=['Intercept'], ax=axs[2], point_estimate=None, hdi_prob='hide', color='purple')

axs[2].axvline(-2, color='k', linestyle='--');

Importantly, we see that while the fixed effects model is focused on recovering the treatment effect on the \(X\) covariate, it does so somewhat at the expense of accuracy on the other systematic components of the model. This focus renders the model less predictively accurate. If we compare the models on the cross-validation score, we see how the hierarchical mundlak model is to be preferred.

compare_df = az.compare({'naive': naive_idata, 'fixed': fixed_effects_idata, 'hierarchical': multilevel_model_idata, 
'mundlak': mundlak_idata})
compare_df

/Users/nathanielforde/mambaforge/envs/pymc_causal/lib/python3.11/site-packages/arviz/stats/stats.py:805: UserWarning: Estimated shape parameter of Pareto distribution is greater than 0.7 for one or more samples. You should consider using a more robust model, this is because importance sampling is less likely to work well if the marginal posterior and LOO posterior are very different. This is more likely to happen with a non-robust model and highly influential observations.
  warnings.warn(

	rank	elpd_loo	p_loo	elpd_diff	weight	se	dse	warning	scale
mundlak	0	-1209.412792	12.298755	0.000000	0.927574	30.226195	0.000000	False	log
hierarchical	1	-1218.354794	27.610000	8.942002	0.000000	30.409392	4.461058	False	log
fixed	2	-1221.023273	33.382146	11.610481	0.044748	31.742948	5.237775	True	log
naive	3	-1295.363481	2.846463	85.950689	0.027678	29.493123	12.984555	False	log

az.plot_compare(compare_df);

This is not the only way to assess viability of the model’s functional form but it’s not a bad way.

Full Luxury Bayesian Mundlak Machine

As good Bayesians we might be worry about the false precision of adding simple point estimates for the group mean covariates in the Mundlak model. We can remedy this by explicitly incorporating these values as an extra parameter and adding uncertainty to the draws on these parameters.

id_indx, unique_ids = pd.factorize(sim_df["g"])

coords = {'ids': list(range(N_groups))}
with pm.Model(coords=coords) as model: 

    x_data = pm.Data('X_data', sim_df['X'])
    z_data = pm.Data('Z_data', sim_df['Z'])
    y_data = pm.Data('Y_data', sim_df['Y'])

    alpha0 = pm.Normal('Intercept', 0, 1)
    alpha_j = pm.Normal('alpha_j', 0, 1, dims='ids')
    beta_xy = pm.Normal('X', 0, 1)
    beta_zy = pm.Normal('Z', 0, 1)

    group_means = pm.Normal('group_means', sim_df.groupby('g')['X'].mean().values, .1, dims='ids')
    group_sigma = pm.HalfNormal('group_sigma', 0.1)
    
    group_mu = pm.Normal('group_mu', (group_means[id_indx]), group_sigma, observed=sim_df['X'].values)

    mu = pm.Deterministic('mu', (alpha0 + alpha_j[id_indx]) + beta_xy*x_data + beta_zy*z_data + group_means[id_indx])
    p = pm.Deterministic("p", pm.math.invlogit(mu))
    # likelihood
    pm.Binomial("y", n=1, p=p, observed=y_data)

    idata = pm.sample(idata_kwargs={"log_likelihood": True})

Instead of merely adding a new feature to a regression we’ve added a second likelihood term for the predictor \(X\) and seek to model this outcome as a function of the group means. Calibrating the group_mean parameter that is also included in our likelihood term for the \(Y\) outcome.

pm.model_to_graphviz(model)

This model bakes more uncertainty into the process assuming a kind of measurement-error model which may be more or less apt depending on how much data you’ve acquired and your view of the underlying process.

az.summary(idata, var_names=['Intercept', 'X', 'Z'])

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
Intercept	-1.873	0.207	-2.231	-1.461	0.007	0.005	1004.0	1691.0	1.0
X	0.989	0.058	0.881	1.100	0.001	0.001	6940.0	2651.0	1.0
Z	-0.518	0.173	-0.842	-0.190	0.005	0.004	1158.0	2069.0	1.0

It also recovers the correct parameter specifcation well.

Recap

We’ve just seen a number of different modelling specifications that attempt to deal with the threat of group level confounding. We’ll now look at a case-study where group level confounding remains highly plausible and the choice of estimation routine is not trivial. This will highlight aspects of how what we seek learn determines our model choice.

Nested Groups, Hierarchies and Fixed Effects

We’ve seen how various attempts to account for the group effects can more or less recover the parameters of a complex data generating process with group confounding. Now we want to look at a case where we can have interacting group effects at multiple levels.

Structure

Pupils within Class Rooms within Schools

A natural three level group hierarchy occurs in the context of educational organisations and business org-charts. We can use this fact to interrogate briefly how inferential statements about treatment effects vary as a function of what and how we control for group level variation. We draw the following data set from Linear Mixed Models: A Practical Guide Using Statistical Software.

df = pd.read_csv('classroom.csv')
df['class_mean'] = df.groupby(['classid'])['mathprep'].transform(np.mean)
df['school_mean'] = df.groupby(['schoolid'])['mathprep'].transform(np.mean)
df.head()

	sex	minority	mathkind	mathgain	ses	yearstea	mathknow	housepov	mathprep	classid	schoolid	childid	class_mean	school_mean
0	1	1	448	32	0.46	1.0	NaN	0.082	2.00	160	1	1	2.00	2.909091
1	0	1	460	109	-0.27	1.0	NaN	0.082	2.00	160	1	2	2.00	2.909091
2	1	1	511	56	-0.03	1.0	NaN	0.082	2.00	160	1	3	2.00	2.909091
3	0	1	449	83	-0.38	2.0	-0.11	0.082	3.25	217	1	4	3.25	2.909091
4	0	1	425	53	-0.03	2.0	-0.11	0.082	3.25	217	1	5	3.25	2.909091

The data has three distinct levels: (1) the child or pupil and their demographic attributes and outcome variable mathgain, (2) the classroom and the teacher level attributes such as their experience yearstea and a record of their mathematics courses taken mathprep, (3) school and neighbourhood level with features describing poverty measures in the vicinity housepov.

We’ll plot the child’s outcome mathgain against the mathprep and distinguish the patterns by school.

Code

def rand_jitter(arr):
    stdev = .01 * (max(arr) - min(arr))
    return arr + np.random.randn(len(arr)) * stdev


schools = df['schoolid'].unique()
schools_10 = [schools[i:i+10] for i in range(0, len(schools), 10)]
fig, axs = plt.subplots(3,4, figsize=(20, 10), 
sharey=True, sharex=True)
axs = axs.flatten()
for s, ax in zip(schools_10, axs):
    temp = df[df['schoolid'].isin(s)]
    ax.scatter(rand_jitter(temp['mathprep']), temp['mathgain'], 
    c=temp['schoolid'], cmap='tab10')
    ax.set_title(f"Schools \n {s}");

There is a small number of observed students per school so the individual school level distributions show some extreme outliers but the overall distribution nicely converges to an approximately normal symmetric shape.

fig, axs = plt.subplots(1, 2, figsize=(10, 6))
axs = axs.flatten()
for school in schools:
    temp = df[df['schoolid'] ==school]
    axs[0].hist(temp['mathgain'], color='grey', alpha=0.3, density=True, histtype='step', cumulative=True)

axs[0].hist(df['mathgain'], bins=30, ec='black', density=True, cumulative=True, histtype='step')
axs[1].hist(df['mathgain'], bins=30, ec='black', density=True, cumulative=False)
axs[0].set_title("Cumulative Distribution Function by School")
axs[1].set_title("Overall Distribution");

With these kinds of structures we need to be careful in how we evaluate any treatment effects when there are reasons to believe in group-level effects that impact the outcome variable. Imagine the true treatment effect is a sprinter running too fast to cleanly measure - each group interaction effect is added to his load as a weight. Enough weights absorb enough of the variation in the treatment that he is dragged to a crawl. Whatever movement he can make while dragging this burden is the effect we attribute to the treatment. Or put another way - the effects of various group interactions will modify the treatment effectiveness in some way and unless we account for this impact in our model, then the inferences regarding the treatment will be clouded.

Interaction Effects and Nuisance Parameters

Not all possible interactions will be present in the data. But if we specify the model to account for various interactions we may explode the number of parameters beyond the number of data points. This can cause issues in estimation routines and requires consideration about what the group parameters are aimed at capturing.

Why add covariates for group-membership when no observation reflects outcomes in that group? We cannot learn anything about these cases. At least if we add the group effects as a hierarchical random effect we induce shrinkage on the parameter estimates towards the mean of the hierarchical parameter in the model. This means that when predicting on “new” data with examples of these missing cases we can predict a sensible default. The distinction rests in the role we have in mind for these tools. If we seriously commit to the idea that group variation reflects a real “common” process in the larger population and we want to learn about that over-arching process then we deploy a random effects model. But if we only see these as tools for accounting to variation in the sample, allowing us to pin down an alternative focal estimate then the group indicator covariates are just “nuisance” parameters and missing cases are irrelevant.

Philosophical Digression

This last point skips a little quickly over a fundamental feature of interpreting these models. If we aim to interpret these models as reflecting a common process across these groups that exists in a “population”, then we’re endorsing an inferential view that extends beyond the sample. We’re actively seeking to learn a general truth about the data generating process which we deem to be adequately expressed in our model. If we seek to “soak up” the variation due to group effects, we’re treating these group effects as noise in the sample data and making inferential commitments only about the focal parameter in the model. This approach to learning differentiates approaches to credible causal inference. On the one hand, fixing your estimand and designing estimators to specifically capture that estimate seems like a modest and compelling strategy. On the other hand if your model ignores aspects of underlying phenomena or fails to retrodict the observable data, it’s dubious as to why anyone would trust its output.

To see the extent of redundany we can examine the dimensions of the covariate matrices that result from including more or less interaction terms.

y, X = dmatrices("mathgain ~ mathprep + C(schoolid)+ C(classid)", df, return_type="dataframe")
print(X.shape)

y, X1 = dmatrices("mathgain ~ mathprep + C(schoolid)/C(classid)", df, return_type="dataframe")
print(X1.shape)


y, X2 = dmatrices("mathgain ~ mathprep + C(schoolid):C(childid)", df, return_type="dataframe")
print(X2.shape)

(1190, 419)

(1190, 33385)

(1190, 127331)

We see here how different ways in which to account for group level variation and interaction effects lead to vastly inflated feature matrices. However not all interaction terms matter, or put another way… nor all the possible interactions feature in the data. So we have likely inflated the data matrix beyond necessity.

Here we define a helper function to parse a complex interaction formula, remove the columns entirely composed of zeros and return a new formula and dataframe which has a suitable range of features to capture the variation structures in the data.

def make_interactions_df(formula, df):
    y, X = dmatrices(formula, df, return_type="dataframe")
    n = X.shape[1]
    X = X[X.columns[~(np.abs(X) < 1e-12).all()]]
    n1 = X.shape[1]
    target_name = y.columns[0]
    d = pd.concat([y, X], axis=1)
    d.drop(['Intercept'], axis=1, inplace=True)
    d.columns = [c.replace('[', '').replace(']','').replace('C(', '').replace(')', '').replace('.', '_').replace(':', '_') for c in d.columns]
    cols = ' + '.join([col for col in d.columns if col != target_name])
    formula = f"{target_name} ~ {cols}"
    print(f"""Size of original interaction features: {n} \nSize of reduced feature set: {n1}""")
    return formula, d

formula, interaction_df = make_interactions_df("mathgain ~ mathprep + C(schoolid):C(childid)", df)

interaction_df.head()

Size of original interaction features: 127331 
Size of reduced feature set: 2370

	mathgain	childidT_2	childidT_3	childidT_4	childidT_5	childidT_6	childidT_7	childidT_8	childidT_9	childidT_10	...	schoolidT_107_childid1182	schoolidT_107_childid1183	schoolidT_107_childid1184	schoolidT_107_childid1185	schoolidT_107_childid1186	schoolidT_107_childid1187	schoolidT_107_childid1188	schoolidT_107_childid1189	schoolidT_107_childid1190	mathprep
0	32.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	...	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	2.00
1	109.0	1.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	...	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	2.00
2	56.0	0.0	1.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	...	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	2.00
3	83.0	0.0	0.0	1.0	0.0	0.0	0.0	0.0	0.0	0.0	...	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	3.25
4	53.0	0.0	0.0	0.0	1.0	0.0	0.0	0.0	0.0	0.0	...	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	3.25

5 rows × 2370 columns

We have reduced the number of interactions by an order of magnitude! We can now fit a regression model to the revised feature matrix.

Comparing Interaction Models

Consider the variation in the coefficient values estimated for mathprep as we add more and more interaction effects. The addition of interaction effects generates a large number of completely 0 interaction terms which we remove here.

formulas = ["""mathgain ~ mathprep + C(schoolid)""",
""" mathgain ~ mathprep + school_mean*class_mean""" , 
"""mathgain ~ mathprep + C(schoolid) + C(classid)""", 
"""mathgain ~ mathprep + C(schoolid)*C(classid)""",
"""mathgain ~ mathprep + C(classid):C(childid)""", 
]

estimates_df = []
for f in formulas:
    formula, interaction_df = make_interactions_df(f, df)
    result = smf.ols(formula, interaction_df).fit()
    estimates = [[result.params['mathprep']], list(result.conf_int().loc['mathprep', :]), [formula]]
    estimates = [e for est in estimates for e in est]
    estimates_df.append(estimates)

estimates_df = pd.DataFrame(estimates_df, columns=['mathprep_estimate', 'lower bound', 'upper bound', 'formula'])

estimates_df

Size of original interaction features: 108 
Size of reduced feature set: 108
Size of original interaction features: 5 
Size of reduced feature set: 5
Size of original interaction features: 419 
Size of reduced feature set: 419

Size of original interaction features: 33385 
Size of reduced feature set: 728

Size of original interaction features: 371281 
Size of reduced feature set: 2376

/Users/nathanielforde/mambaforge/envs/pymc_causal/lib/python3.11/site-packages/statsmodels/regression/linear_model.py:1717: RuntimeWarning: divide by zero encountered in scalar divide
  return np.dot(wresid, wresid) / self.df_resid

	mathprep_estimate	lower bound	upper bound	formula
0	1.060768	-1.435118	3.556655	mathgain ~ schoolidT_2 + schoolidT_3 + schooli...
1	1.789413	-1.587051	5.165876	mathgain ~ mathprep + school_mean + class_mean...
2	2.948546	0.601150	5.295942	mathgain ~ schoolidT_2 + schoolidT_3 + schooli...
3	3.545931	1.359793	5.732068	mathgain ~ schoolidT_2 + schoolidT_3 + schooli...
4	2.303187	NaN	NaN	mathgain ~ childidT_2 + childidT_3 + childidT_...

The point here (perhap obvious) is that the estimate of treatment effects due to some policy or programme can be differently understood when the regression model is able to account for increasing aspects of individual variation. Choice of the right way to “saturate” your regression specification are at the heart of causal inference. We will consider a number of specifications below that incorporate these group effects in a hierarchical model which nests the effect of class-membership within school membership. This choice allows us to control for group specific interactions without worrying about over-indexing on the observed interaction effects in the sample data requiring that we handle more fixed effects parameters than we have data points.

In what follows we’ll specify a nested approach to the parameter specifcation using a random effects model. The idea here is that classes are already implicitly nested in schools and so we don’t need to add parameters for classes at multiple schools. Additionally we’re positing that there is independent interest in the effectiveness school/class effects i.e. the degree to which variation in a school/class nest can account for variation in the outcome.

Minimal Model

model = bmb.Model(f"mathgain ~ mathprep + (1 | schoolid / classid)", df)
idata = model.fit( inference_method="nuts_numpyro",
    idata_kwargs={"log_likelihood": True})

The model specification here is deliberately minimalist we want to observe how much of the variation in the outcome can be accounted for by solely adding extensive controls for interactions of group level effects and the treatment but ignoring all else.

model.graph()

We can see the derived sigma parameters here which can be understood as partialling out the variance of the outcome into components due to those group level effects and the unexplained residuals.

az.summary(idata, var_names=['Intercept', '1|schoolid_sigma', '1|schoolid:classid_sigma', 'mathgain_sigma', 'mathprep'])

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
Intercept	52.409	3.511	45.811	58.926	0.068	0.048	2667.0	2897.0	1.00
1|schoolid_sigma	8.502	2.231	4.425	12.557	0.140	0.099	373.0	237.0	1.01
1|schoolid:classid_sigma	9.885	2.249	5.648	14.035	0.094	0.068	561.0	906.0	1.01
mathgain_sigma	32.123	0.766	30.589	33.484	0.017	0.012	2000.0	2418.0	1.00
mathprep	1.931	1.241	-0.297	4.295	0.024	0.017	2713.0	2782.0	1.00

Note here the relative proportion of the school specifc variances 1|schoolid_sigma to the overall variance of the residuals mathgain_sigma.

Calculating the IntraClass Correlation Coefficient

These models faciliate the calculation of the ICC statistics which is a measure of “explained variance”. The thought is to gauge the proportion of variance ascribed to one set of random effects over and above the total estimated variance in the baseline model, including the residuals mathgain_sigma.

a = idata['posterior']['1|schoolid_sigma']**2

b = (idata['posterior']['1|schoolid:classid_sigma']**2 + idata['posterior']['1|schoolid_sigma']**2)

c = (idata['posterior']['1|schoolid:classid_sigma']**2 + idata['posterior']['1|schoolid_sigma']**2 + idata['posterior']['mathgain_sigma']**2)

(a / c).mean().item() 

0.06340858486893981

((a + b) / c).mean().item()

0.21140567611242062

We can see here that the interaction terms do seem to account for a goodly portion of the variance in the outcome and we ought to consider retaining their inclusion in our modelling work. The structure of the problem drives us towards their inclusion. Class/school effects are going to absorb a sufficient portion of the variation. So they merit study in their own right, lest the individual class/school dynamics obscure the effectiveness of the mathprep treatment. Similarly, it’s likely valuable to consider the efficacy of the average class/school in a wider policy conversation.

Augmenting the Models

Next we augment our model with more pupil level control variables aiming to pin down some of the aspects of the variation in the outcome.

Adding Pupil Fixed Effects

Here we add these fixed effects population parameters. But note they are not merely devices for controlling variance in the outcome, they’re interpretation is likely of independent interest.

model_fixed = bmb.Model(f"mathgain ~ mathkind + sex + minority + ses + mathprep + (1 | schoolid / classid)", df)
idata_fixed = model_fixed.fit( inference_method="nuts_numpyro",
    idata_kwargs={"log_likelihood": True})

az.summary(idata_fixed, var_names=['Intercept', 
'mathkind', 'sex', 'minority', 'ses', 'mathprep',
'1|schoolid_sigma', '1|schoolid:classid_sigma', 'mathgain_sigma'])

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
Intercept	280.431	11.376	259.060	301.382	0.152	0.107	5599.0	3135.0	1.00
mathkind	-0.469	0.022	-0.511	-0.428	0.000	0.000	5424.0	3247.0	1.00
sex	-1.243	1.696	-4.468	1.864	0.022	0.021	5933.0	2990.0	1.00
minority	-8.161	2.411	-12.510	-3.245	0.037	0.027	4138.0	2816.0	1.00
ses	5.401	1.253	3.164	7.822	0.017	0.013	5288.0	2775.0	1.00
mathprep	0.779	1.058	-1.172	2.846	0.017	0.013	4120.0	3105.0	1.00
1|schoolid_sigma	8.665	1.595	5.807	11.788	0.052	0.037	975.0	1152.0	1.01
1|schoolid:classid_sigma	9.100	1.802	5.827	12.639	0.081	0.057	533.0	656.0	1.01
mathgain_sigma	27.167	0.657	25.868	28.333	0.015	0.010	2001.0	2400.0	1.00

Now we add a further class level control.

Adding Class Level Fixed Effects

model_fixed_1 = bmb.Model(f"mathgain ~ mathkind + sex + minority + ses + yearstea + mathknow + mathprep + (1 | schoolid / classid)", df.dropna())
idata_fixed_1 = model_fixed_1.fit( inference_method="nuts_numpyro",
    idata_kwargs={"log_likelihood": True})

az.summary(idata_fixed_1, var_names=['Intercept', 
'mathkind', 'sex', 'minority', 'ses', 'yearstea', 'mathknow', 'mathprep','1|schoolid_sigma', '1|schoolid:classid_sigma', 'mathgain_sigma'])

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
Intercept	281.844	11.998	259.508	304.613	0.202	0.143	3523.0	2961.0	1.0
mathkind	-0.475	0.023	-0.518	-0.431	0.000	0.000	3753.0	2825.0	1.0
sex	-1.343	1.742	-4.551	1.872	0.025	0.022	4691.0	2659.0	1.0
minority	-7.919	2.446	-12.646	-3.566	0.044	0.032	3121.0	3013.0	1.0
ses	5.439	1.256	3.184	7.868	0.020	0.015	3853.0	2830.0	1.0
yearstea	0.038	0.120	-0.190	0.253	0.002	0.002	2538.0	2625.0	1.0
mathknow	1.882	1.185	-0.401	4.081	0.024	0.017	2385.0	2641.0	1.0
mathprep	1.094	1.148	-1.079	3.185	0.023	0.017	2507.0	2732.0	1.0
1|schoolid_sigma	8.668	1.620	5.704	11.771	0.056	0.040	857.0	1060.0	1.0
1|schoolid:classid_sigma	9.219	1.812	5.574	12.323	0.072	0.051	663.0	759.0	1.0
mathgain_sigma	26.776	0.651	25.562	28.008	0.014	0.010	2095.0	2616.0	1.0

fig, ax = plt.subplots(figsize=(10, 7))
az.plot_forest([idata, idata_fixed, idata_fixed_1], combined=True, var_names=['mathprep', '1|schoolid_sigma', '1|schoolid:classid_sigma', 'mathgain_sigma'], ax=ax)
ax.axvline(1)

<matplotlib.lines.Line2D at 0x2f44aed10>

We now make use of bambis model interpretation module to plot the marginal effect on the outcome due to changes in the treatment intensity.

Code

fig, axs = plt.subplots(3, 1, figsize=(10, 14), 
dpi=120, sharey=True, sharex=True)
axs = axs.flatten()
axs[0].axhline(50, color='red', label='Reference Line', 
linestyle='--')
axs[1].axhline(50, color='red', label='Reference Line', 
linestyle='--')
axs[2].axhline(50, color='red', label='Reference Line', 
linestyle='--')
bmb.interpret.plot_predictions(model, idata, "mathprep", ax=axs[0]);
bmb.interpret.plot_predictions(model_fixed, idata_fixed, "mathprep", ax=axs[1]);
bmb.interpret.plot_predictions(model_fixed_1, idata_fixed_1, "mathprep", ax=axs[2]);
axs[0].set_title("Variation in Implied Outcome by Treatment \n mathgain ~ mathprep + (1 | schoolid / classid) ")
axs[1].set_title("Variation in Implied Outcome by Treatment \n mathgain ~ mathkind + sex + minority + ses + mathprep + (1 | schoolid / classid) ")
axs[2].set_title("Variation in Implied Outcome by Treatment \n mathgain ~ mathkind + sex + minority + ses + yearstea + \n mathknow + mathprep + (1 | schoolid / classid")
axs[0].set_xlabel('')
axs[1].set_xlabel('')
axs[0].legend();

As we can see here across all the different model specifications we see quite modest effects of treatment with very wide uncertainty. So far, so what?! You might be sceptical that teacher training has any real discernible impact on child outcomes? Maybe you believe other interventions are more important to fund? These kinds of questions determine policy. So misguided policy interventions on child-hood education can have radical consequences.

It’s, therefore, vital that we have robust and justifiable approaches to the analysis of these policy questions in the face of group level confounding. This last point is crucial - when model complexity balloons due extensive interaction effects, then we need efficient tools to interrogate the outcome level differences implied by the model choices. Your intuition and understanding of a process is keyed to the observable effects of that process. Your understanding of the internal mechanics of a model is likely less than your concrete expectations of observable behaviour. As such, credibility requires that we be able to assess models on the variation produce in the outcome variable rather merely on the parameters values.

Two Way Fixed Effects and Temporal Confounding

Difference in Differences designs are the overworked donkeys of social science. Many, many studies stand or fall by the assumptions baked into DiD designs. There are at least two aspects to these assumptions (i) the substantive commitments about the data generating process (e.g. parrallel trends) and (ii) the appropriateness of the functional form used to model (i). We will look first at a case where all the assumptions can be met, and then examine how things break-down under challenging staggered treatment regimes.

Event Studies and Change in Time

We take this panel data set from the pyfixest package.

url = "https://raw.githubusercontent.com/py-econometrics/pyfixest/master/pyfixest/did/data/df_het.csv"
df_het = pd.read_csv(url)
df_het.head()

	unit	state	group	unit_fe	g	year	year_fe	treat	rel_year	rel_year_binned	error	te	te_dynamic	dep_var
0	1	33	Group 2	7.043016	2010	1990	0.066159	False	-20.0	-6	-0.086466	0	0.0	7.022709
1	1	33	Group 2	7.043016	2010	1991	-0.030980	False	-19.0	-6	0.766593	0	0.0	7.778628
2	1	33	Group 2	7.043016	2010	1992	-0.119607	False	-18.0	-6	1.512968	0	0.0	8.436377
3	1	33	Group 2	7.043016	2010	1993	0.126321	False	-17.0	-6	0.021870	0	0.0	7.191207
4	1	33	Group 2	7.043016	2010	1994	-0.106921	False	-16.0	-6	-0.017603	0	0.0	6.918492

Panel data of this kind of structure is difficult to intuit unless visualised. The key structures on which this DiD estimator works are the averages across state and time. We depict these here:

Code

fig, axs = plt.subplots(2, 1, figsize=(10, 12))
axs = axs.flatten()
for u in df_het['unit'].unique():
    temp = df_het[df_het['unit']==u]
    axs[0].plot(temp['year'], temp['dep_var'], color='grey', alpha=0.01)
    axs[1].plot(temp['year'], temp['dep_var'], color='grey', alpha=0.01)
df_het.groupby(['year', 'state'])[['dep_var']].mean().reset_index().pivot(index='year', columns='state', values='dep_var').plot(ax=axs[0], legend=False, color='blue', 
alpha=0.4)

df_het.groupby(['year', 'g'])[['dep_var']].mean().reset_index().pivot(index='year', columns='g', values='dep_var').plot(ax=axs[1], legend=False)

axs[0].set_title("Difference in Differences \n State Mean Change an Individual Trajectories")
axs[1].set_title("Difference in Differences \n Mean Change an Individual Trajectories");

Note how the blue line represents a cohort that never undergoes a treatment and is maintained as a coherent control group throughout the sequence even though we have two other cohorts. This ensures we have a constant contrast case allowing us to identify the causal estimand of interest. It’s also worth calling out that we have heterogenous patterns in the treatment effects across the cohorts i.e. the the intensity of the treatment effect differs across the cohorts.

TWFE in pyfixest

A natural way to interrogate these kinds of question is to wonder how the treatment effect evolves over time? Is it initially impactful converging to a quick plateau or a slowly building pattern of consistent growth?

fit_twfe_event = pf.feols(
    "dep_var ~ i(rel_year, ref=-1.0) | state + year ",
    df_het,
    vcov={"CRV1": "state"},
)

fit_twfe_event.tidy()

OMP: Info #276: omp_set_nested routine deprecated, please use omp_set_max_active_levels instead.

	Estimate	Std. Error	t value	Pr(>|t|)	2.5%	97.5%
Coefficient
C(rel_year, contr.treatment(base=-1.0))[T.-20.0]	-0.099445	0.081699	-1.217202	0.230842	-0.264697	0.065808
C(rel_year, contr.treatment(base=-1.0))[T.-19.0]	0.000624	0.083213	0.007493	0.994060	-0.167690	0.168937
C(rel_year, contr.treatment(base=-1.0))[T.-18.0]	0.004125	0.089719	0.045974	0.963565	-0.177349	0.185599
C(rel_year, contr.treatment(base=-1.0))[T.-17.0]	0.021899	0.085686	0.255573	0.799624	-0.151418	0.195216
C(rel_year, contr.treatment(base=-1.0))[T.-16.0]	-0.036933	0.096921	-0.381067	0.705221	-0.232975	0.159108
C(rel_year, contr.treatment(base=-1.0))[T.-15.0]	0.069578	0.081289	0.855936	0.397262	-0.094844	0.234000
C(rel_year, contr.treatment(base=-1.0))[T.-14.0]	0.037734	0.086618	0.435641	0.665499	-0.137467	0.212936
C(rel_year, contr.treatment(base=-1.0))[T.-13.0]	0.061779	0.083362	0.741098	0.463073	-0.106836	0.230394
C(rel_year, contr.treatment(base=-1.0))[T.-12.0]	0.089913	0.084900	1.059044	0.296095	-0.081814	0.261639
C(rel_year, contr.treatment(base=-1.0))[T.-11.0]	0.000982	0.079104	0.012417	0.990156	-0.159021	0.160986
C(rel_year, contr.treatment(base=-1.0))[T.-10.0]	-0.113033	0.064922	-1.741047	0.089560	-0.244351	0.018285
C(rel_year, contr.treatment(base=-1.0))[T.-9.0]	-0.069225	0.057046	-1.213481	0.232244	-0.184612	0.046162
C(rel_year, contr.treatment(base=-1.0))[T.-8.0]	-0.061290	0.060362	-1.015370	0.316188	-0.183383	0.060804
C(rel_year, contr.treatment(base=-1.0))[T.-7.0]	-0.002022	0.064602	-0.031306	0.975185	-0.132693	0.128648
C(rel_year, contr.treatment(base=-1.0))[T.-6.0]	-0.055810	0.064674	-0.862938	0.393447	-0.186626	0.075006
C(rel_year, contr.treatment(base=-1.0))[T.-5.0]	-0.065009	0.064810	-1.003066	0.322012	-0.196099	0.066082
C(rel_year, contr.treatment(base=-1.0))[T.-4.0]	-0.009850	0.053098	-0.185505	0.853794	-0.117251	0.097551
C(rel_year, contr.treatment(base=-1.0))[T.-3.0]	0.046338	0.062950	0.736104	0.466072	-0.080991	0.173667
C(rel_year, contr.treatment(base=-1.0))[T.-2.0]	0.015947	0.065442	0.243687	0.808751	-0.116421	0.148316
C(rel_year, contr.treatment(base=-1.0))[T.0.0]	1.406404	0.055641	25.276422	0.000000	1.293860	1.518949
C(rel_year, contr.treatment(base=-1.0))[T.1.0]	1.660552	0.065033	25.533859	0.000000	1.529010	1.792094
C(rel_year, contr.treatment(base=-1.0))[T.2.0]	1.728790	0.056793	30.440100	0.000000	1.613915	1.843665
C(rel_year, contr.treatment(base=-1.0))[T.3.0]	1.854839	0.058315	31.807288	0.000000	1.736886	1.972792
C(rel_year, contr.treatment(base=-1.0))[T.4.0]	1.958676	0.071144	27.531232	0.000000	1.814774	2.102578
C(rel_year, contr.treatment(base=-1.0))[T.5.0]	2.082161	0.063855	32.607651	0.000000	1.953002	2.211320
C(rel_year, contr.treatment(base=-1.0))[T.6.0]	2.191062	0.068510	31.981462	0.000000	2.052487	2.329637
C(rel_year, contr.treatment(base=-1.0))[T.7.0]	2.279073	0.075014	30.381921	0.000000	2.127342	2.430803
C(rel_year, contr.treatment(base=-1.0))[T.8.0]	2.364593	0.058598	40.352477	0.000000	2.246067	2.483120
C(rel_year, contr.treatment(base=-1.0))[T.9.0]	2.372163	0.056560	41.940696	0.000000	2.257759	2.486566
C(rel_year, contr.treatment(base=-1.0))[T.10.0]	2.649271	0.056177	47.158942	0.000000	2.535641	2.762901
C(rel_year, contr.treatment(base=-1.0))[T.11.0]	2.753591	0.075526	36.458951	0.000000	2.600826	2.906356
C(rel_year, contr.treatment(base=-1.0))[T.12.0]	2.813935	0.079320	35.475900	0.000000	2.653496	2.974375
C(rel_year, contr.treatment(base=-1.0))[T.13.0]	2.756070	0.078368	35.168251	0.000000	2.597555	2.914584
C(rel_year, contr.treatment(base=-1.0))[T.14.0]	2.863427	0.098389	29.103072	0.000000	2.664416	3.062438
C(rel_year, contr.treatment(base=-1.0))[T.15.0]	2.986652	0.093309	32.008066	0.000000	2.797916	3.175388
C(rel_year, contr.treatment(base=-1.0))[T.16.0]	2.963032	0.085427	34.684870	0.000000	2.790239	3.135825
C(rel_year, contr.treatment(base=-1.0))[T.17.0]	2.972596	0.092390	32.174553	0.000000	2.785721	3.159472
C(rel_year, contr.treatment(base=-1.0))[T.18.0]	2.935051	0.094126	31.182203	0.000000	2.744664	3.125439
C(rel_year, contr.treatment(base=-1.0))[T.19.0]	2.918707	0.084193	34.667036	0.000000	2.748411	3.089002
C(rel_year, contr.treatment(base=-1.0))[T.20.0]	2.979701	0.087777	33.946394	0.000000	2.802156	3.157246
C(rel_year, contr.treatment(base=-1.0))[T.inf]	0.128053	0.078513	1.630976	0.110946	-0.030755	0.286861

The model specification defines indicator variables for the relative years before and after the penultimate year before treatment is applied, it then incorporates the fixed effects for state and year. The coefficient values ascribed the the relative year indicators are used to plot the event-study trajectories. This is a two-way fixed effects estimation routine where the fixed effects for state and year indicators absorb the variance due to those groupings. It is often estimated using a de-meaning technique which we will demonstrate below.

figsize = [1200, 600]
fit_twfe_event.iplot(
    coord_flip=False,
    title="TWFE-Estimator",
    figsize=figsize,
    xintercept=18.5,
    yintercept=0,
).show()

We can also aim to marginalise over the details of the state trajectories by defining the similar estimation routine on the individuals and their treatment indicator.

fit_twfe = pf.feols(
    "dep_var ~ i(treat) | unit + year",
    df_het,
    vcov={"CRV1": "state"},
)

fit_twfe.tidy()

	Estimate	Std. Error	t value	Pr(>|t|)	2.5%	97.5%
Coefficient
C(treat)[T.True]	1.98254	0.019331	102.55618	0.0	1.943439	2.021642

De-meaning and TWFEs

We’ve seen above that the fixed-effect estimators in these DiD designs involve a lot of indicator variables. These are largely not the focus on the question at hand but are used exlusively to absorb the noise that takes away from our understanding of the treatment effect. We can achieve similar results with less parameters required if we “de-mean” the focus variables by the group averages of the control factors of state and year or unit. This operation makes for more efficient TWFE estimation routines is provably a variety of mundlak regression as shown by Woolridge 2021.

We implement the de-meaning technique and show “equivalence by python” as follows:

def demean(df, col_to_demean, group):
    return df.assign(**{col_to_demean: (df[col_to_demean]
                                        - df.groupby(group)[col_to_demean].transform("mean")
                                        )})


def apply_demeaning(df_het, by=['state', 'year'], event=True):
    if event: 
        d = pd.get_dummies(df_het['rel_year']).drop(-1, axis=1) 
        d.columns = ['rel_year_' +str(c).replace('-', 'minus_') for c in d.columns]
    else:
        d = df_het[['treat']]
    d[by[0]] = df_het[by[0]]
    d[by[1]] = df_het[by[1]]
    for col in d.columns: 
        if col in by:
            pass
        else: 
            for c in by:
                d = demean(d, col, c)
    d = d.drop(by, axis=1)
    d['dep_var'] = df_het['dep_var']
    return d
## Demean the relative years features for event studies
d_event = apply_demeaning(df_het, by=['state', 'year'], event=True)

## Demean the treatment indicator for ATT estimation
d = apply_demeaning(df_het, by=['unit', 'year'], event=False)

d_event.head()

	rel_year_minus_20.0	rel_year_minus_19.0	rel_year_minus_18.0	rel_year_minus_17.0	rel_year_minus_16.0	rel_year_minus_15.0	rel_year_minus_14.0	rel_year_minus_13.0	rel_year_minus_12.0	rel_year_minus_11.0	...	rel_year_13.0	rel_year_14.0	rel_year_15.0	rel_year_16.0	rel_year_17.0	rel_year_18.0	rel_year_19.0	rel_year_20.0	rel_year_inf	dep_var
0	0.651694	-0.002973	-0.002973	-0.002973	-0.002973	-0.002973	-0.002973	-0.002973	-0.002973	-0.002973	...	0.000734	0.000734	0.000734	0.000734	0.000734	0.000734	0.000734	0.000734	-0.28125	7.022709
1	-0.002973	0.651694	-0.002973	-0.002973	-0.002973	-0.002973	-0.002973	-0.002973	-0.002973	-0.002973	...	0.000734	0.000734	0.000734	0.000734	0.000734	0.000734	0.000734	0.000734	-0.28125	7.778628
2	-0.002973	-0.002973	0.651694	-0.002973	-0.002973	-0.002973	-0.002973	-0.002973	-0.002973	-0.002973	...	0.000734	0.000734	0.000734	0.000734	0.000734	0.000734	0.000734	0.000734	-0.28125	8.436377
3	-0.002973	-0.002973	-0.002973	0.651694	-0.002973	-0.002973	-0.002973	-0.002973	-0.002973	-0.002973	...	0.000734	0.000734	0.000734	0.000734	0.000734	0.000734	0.000734	0.000734	-0.28125	7.191207
4	-0.002973	-0.002973	-0.002973	-0.002973	0.651694	-0.002973	-0.002973	-0.002973	-0.002973	-0.002973	...	0.000734	0.000734	0.000734	0.000734	0.000734	0.000734	0.000734	0.000734	-0.28125	6.918492

5 rows × 42 columns

We now have a data set with 42 columns focused on the treatment structures, but implicitly controls for the variation due to state and time. We’ll see below that this representation of the data will correctly estimate the treatment effects.

Event Study and De-Meaning

Now we’ll use the de-meaned data structure above to estimate an event study using Bambi.

x_cols = ' + '.join([c for c in d_event.columns if c != 'dep_var'])
model_twfe_event = bmb.Model(f"dep_var ~ + {x_cols}", d_event)
idata_twfe_event = model_twfe_event.fit( inference_method="nuts_numpyro",
    idata_kwargs={"log_likelihood": True},)

model_twfe_event

       Formula: dep_var ~ + rel_year_minus_20.0 + rel_year_minus_19.0 + rel_year_minus_18.0 + rel_year_minus_17.0 + rel_year_minus_16.0 + rel_year_minus_15.0 + rel_year_minus_14.0 + rel_year_minus_13.0 + rel_year_minus_12.0 + rel_year_minus_11.0 + rel_year_minus_10.0 + rel_year_minus_9.0 + rel_year_minus_8.0 + rel_year_minus_7.0 + rel_year_minus_6.0 + rel_year_minus_5.0 + rel_year_minus_4.0 + rel_year_minus_3.0 + rel_year_minus_2.0 + rel_year_0.0 + rel_year_1.0 + rel_year_2.0 + rel_year_3.0 + rel_year_4.0 + rel_year_5.0 + rel_year_6.0 + rel_year_7.0 + rel_year_8.0 + rel_year_9.0 + rel_year_10.0 + rel_year_11.0 + rel_year_12.0 + rel_year_13.0 + rel_year_14.0 + rel_year_15.0 + rel_year_16.0 + rel_year_17.0 + rel_year_18.0 + rel_year_19.0 + rel_year_20.0 + rel_year_inf
        Family: gaussian
          Link: mu = identity
  Observations: 46500
        Priors: 
    target = mu
        Common-level effects
            Intercept ~ Normal(mu: 4.7062, sigma: 7.1549)
            rel_year_minus_20.0 ~ Normal(mu: 0.0, sigma: 83.8218)
            rel_year_minus_19.0 ~ Normal(mu: 0.0, sigma: 83.8218)
            rel_year_minus_18.0 ~ Normal(mu: 0.0, sigma: 83.8218)
            rel_year_minus_17.0 ~ Normal(mu: 0.0, sigma: 83.8218)
            rel_year_minus_16.0 ~ Normal(mu: 0.0, sigma: 83.8218)
            rel_year_minus_15.0 ~ Normal(mu: 0.0, sigma: 83.8218)
            rel_year_minus_14.0 ~ Normal(mu: 0.0, sigma: 83.8218)
            rel_year_minus_13.0 ~ Normal(mu: 0.0, sigma: 83.8218)
            rel_year_minus_12.0 ~ Normal(mu: 0.0, sigma: 83.8218)
            rel_year_minus_11.0 ~ Normal(mu: 0.0, sigma: 83.8218)
            rel_year_minus_10.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_minus_9.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_minus_8.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_minus_7.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_minus_6.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_minus_5.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_minus_4.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_minus_3.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_minus_2.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_0.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_1.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_2.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_3.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_4.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_5.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_6.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_7.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_8.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_9.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_10.0 ~ Normal(mu: 0.0, sigma: 60.2319)
            rel_year_11.0 ~ Normal(mu: 0.0, sigma: 86.6805)
            rel_year_12.0 ~ Normal(mu: 0.0, sigma: 86.6805)
            rel_year_13.0 ~ Normal(mu: 0.0, sigma: 86.6805)
            rel_year_14.0 ~ Normal(mu: 0.0, sigma: 86.6805)
            rel_year_15.0 ~ Normal(mu: 0.0, sigma: 86.6805)
            rel_year_16.0 ~ Normal(mu: 0.0, sigma: 86.6805)
            rel_year_17.0 ~ Normal(mu: 0.0, sigma: 86.6805)
            rel_year_18.0 ~ Normal(mu: 0.0, sigma: 86.6805)
            rel_year_19.0 ~ Normal(mu: 0.0, sigma: 86.6805)
            rel_year_20.0 ~ Normal(mu: 0.0, sigma: 86.6805)
            rel_year_inf ~ Normal(mu: 0.0, sigma: 15.2312)
        
        Auxiliary parameters
            sigma ~ HalfStudentT(nu: 4.0, sigma: 2.862)
------
* To see a plot of the priors call the .plot_priors() method.
* To see a summary or plot of the posterior pass the object returned by .fit() to az.summary() or az.plot_trace()

We can then plot the event-study and observe a similar pattern to the one observed with pyfixest.

def plot_event_study(idata, ax, color='blue', model='demeaned'):
    summary_df = az.summary(idata)
    cols = [i for i in summary_df.index if 'rel' in i]
    summary_df = summary_df[summary_df.index.isin(cols)]
    x = range(len(summary_df))
    ax.scatter(x, summary_df['mean'], label=model, color=color)
    ax.plot([x, x], [summary_df['hdi_3%'],summary_df['hdi_97%']],   color=color)
    ax.set_title("Event Study", fontsize=20)
    return ax

fig, ax = plt.subplots(figsize=(10, 7))
plot_event_study(idata_twfe_event, ax)
ax.legend();

Similarly, we can de-mean the simple treatment indicator using the group means and marginalise over time periods to find a single treatment effect estimate.

model_twfe_trt_demean = bmb.Model(f"dep_var ~ treat", d)
idata_twfe_trt_demean = model_twfe_trt_demean.fit( inference_method="nuts_numpyro",
    idata_kwargs={"log_likelihood": True},)

az.summary(idata_twfe_trt_demean)

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
Intercept	4.706	0.013	4.681	4.730	0.000	0.000	3519.0	2701.0	1.0
treat	1.983	0.049	1.890	2.072	0.001	0.001	3219.0	2804.0	1.0
dep_var_sigma	2.811	0.009	2.793	2.828	0.000	0.000	5914.0	3004.0	1.0

Which again accords with the reported values from pyfixest. This is equivalent to using a Mundlak device as we can see below:

TWFE by Mundlak Device

Woolridge recounts that the TWFE is equivalent to a Mundlak regression where:

[The Mundlak devices] \(\bar{X_{i}}\) and \(\bar{X_{t}}\) effectively act as sufficient statistics in accounting for any unit-specific heterogeneity and time-specific heterogeneity that is correlated with \(X_{it}\). Rather than having to include (N − 1) + (T − 1) control variables, it suffices to include 2K control variables, \(\bar{X_{i}}\), \(\bar{X_{t}}\) - Two-Way Fixed Effects, the Two-Way Mundlak Regression, and Difference-in-Differences Estimators

We will see an example of this equivalence here.

df_het['unit_mean'] = df_het.groupby('unit')['treat'].transform(np.mean)
df_het['time_mean'] = df_het.groupby('year')['treat'].transform(np.mean)

model_twfe_trt = bmb.Model(f"dep_var ~ treat", df_het)
idata_twfe_trt = model_twfe_trt.fit( inference_method="nuts_numpyro",
    idata_kwargs={"log_likelihood": True},)

model_twfe_trt_mundlak = bmb.Model(f"dep_var ~ treat + unit_mean + time_mean", df_het)
idata_twfe_trt_mundlak = model_twfe_trt_mundlak.fit( inference_method="nuts_numpyro",
    idata_kwargs={"log_likelihood": True},)

az.plot_forest([idata_twfe_trt_demean, idata_twfe_trt_mundlak, idata_twfe_trt], combined=True, var_names=['treat'], model_names=['De-meaned', 'Mundlak', 'Simple']);

The de-meaned TWFE estimator and the Mundlak specification result in identical estimates and differ from the naive estimate that fails to control group level confounds. The Mundlak specification is far easier to implement.

Functional Form and Saturated Regression

The vanilla TWFE estimator can successfully recover the treatment effects in DiD designs and facilitate event studies of the same. However, the details of the estimation matter because this functional form is not always robust. Here we’ll see other options that can recover substantially the same inferences and may prove more robust as we’ll see below. The key to each is to articulate enough structural features that allow the model to modify effects based on the suspected group level confounds.

df_het['state_mean'] = df_het.groupby('state')['treat'].transform(np.mean)
df_het['time_mean'] = df_het.groupby('year')['treat'].transform(np.mean)
df_het['cohort_mean'] = df_het.groupby('group')['treat'].transform(np.mean)

model_twfe_event_1 = bmb.Model(f"dep_var ~ 1 + C(year) + state_mean + C(rel_year, Treatment(reference=-1)) ", df_het)
idata_twfe_event_1 = model_twfe_event_1.fit( inference_method="nuts_numpyro",
    idata_kwargs={"log_likelihood": True})


formula = """ dep_var ~ 1 + time_mean:state_mean + C(rel_year, Treatment(reference=-1))"""
twfe_model_ols = smf.ols(formula, data=df_het).fit()
twfe_model_ols.summary()
param_est = pd.DataFrame(twfe_model_ols.params, columns=['estimate']).iloc[1:-1]
param_est['index_number'] = list(range(len(param_est)))
temp = (param_est.reset_index()
)
param_est = temp[(~temp['index'].str.contains(':')) & (temp['index'].str.contains('rel'))]
param_est.reset_index(inplace=True)


model_twfe_event_2 = bmb.Model(f"dep_var ~ (1 | year) + state_mean + C(rel_year, Treatment(reference=-1)) ", df_het)
idata_twfe_event_2 = model_twfe_event_2.fit( inference_method="nuts_numpyro",
    idata_kwargs={"log_likelihood": True})

Having estimated the various alternatives model specifications we compare each against our baseline de-meaned event-study.

Code

fig, axs = plt.subplots(2, 2, figsize=(20, 10))
axs = axs.flatten()
plot_event_study(idata_twfe_event, axs[0], model='Manual DeMeaned')
plot_event_study(idata_twfe_event, axs[1], model='Manual DeMeaned')
plot_event_study(idata_twfe_event, axs[2], model='Manual DeMeaned')
plot_event_study(idata_twfe_event, axs[3], model='Manual DeMeaned')
plot_event_study(idata_twfe_event_1, axs[0], color='green', model='Fixed Effects Saturated Bayes')
plot_event_study(idata_twfe_event_2, axs[1], color='purple', model='Hierarchical Effects Saturated Bayes')
axs[2].scatter(param_est['index'], param_est['estimate'], color='red', label='Mundlak Interaction Features OLS')
tidy = fit_twfe_event.tidy()
xs = range(len(tidy))
tidy.reset_index(inplace=True)
axs[3].scatter(xs, tidy['Estimate'], color='orange', label='pyfixest TWFE')
axs[3].plot([xs, xs], [tidy['2.5%'],tidy['97.5%']], color='orange')
axs[2].set_xticks([])
axs[0].set_title("dep_var ~ 1 + C(year) + state_mean + C(rel_year, Treatment(reference=-1))")
axs[1].set_title("dep_var ~ (1 | year) + state_mean + C(rel_year, Treatment(reference=-1))")
axs[2].set_title("dep_var ~ 1 + time_mean:state_mean + C(rel_year, Treatment(reference=-1))")
axs[3].set_title("dep_var ~ i(rel_year, ref=-1.0) | state + year")
axs[0].legend()
axs[1].legend()
axs[2].legend()
axs[3].legend();

`

This suggests that there are a variety of functional forms even just using regression specifications that seek to control from different types of group level confounding. In this example data most of the functional forms that seek to control for time and state level effects seem to converge in their answers. We will now switch to an example where the vanilla TWFE breaks down.

Issues with TWFE and Richly Parameterised Linear Models

We draw on the example from Pedro Sant’Anna here where it is demonstrated that the vanilla TWFE estimator breaks down under various conditions. These conditions are often related to staggered roll out of a treatment and thereby induces dynamic changes in the treatment group over time. Appropriate inference needs to carefully control for the interaction effects due to staggered treatment.

Let’s generate some data.

true_mu = 1

def make_data(nobs = 1000, nstates = 40):
    ids = list(range(nobs))
    states = np.random.choice(range(nstates), size=nobs, replace=True)
    unit = pd.DataFrame({'unit': ids, 
                        'state': states, 
                        'unit_fe': np.random.normal(states/5, 1, size=nobs),
                        'mu': true_mu})
    
    year = pd.DataFrame({'year': pd.date_range('1980-01-01', '2010-01-01', freq='y'), 
    'year_fe': np.random.normal(0, 1, 30) })
    year['year'] = year['year'].dt.year

    treat_taus = pd.DataFrame({'state': np.random.choice(range(nstates), size=nstates, replace=False),
    'cohort_year': np.sort([1986, 1992, 1998, 2004]*10)
    })

    cross_join = pd.DataFrame([row for row in product(range(nobs), year['year'].unique())], columns =['unit', 'year'])
    cross_join = cross_join.merge(unit, how='left', left_on='unit', 
    right_on='unit')
    cross_join = cross_join.merge(year, how='left', left_on='year', 
    right_on='year')
    cross_join = cross_join.merge(treat_taus, how='left', left_on='state', right_on='state')
    cross_join = cross_join.assign(
        error = np.random.normal(0, 1, len(cross_join)),
        treat = lambda x: np.where(x['year'] >= x['cohort_year'], 1, 0)
    )
    cross_join = cross_join.assign(tau = np.where(cross_join['treat'] == 1, cross_join['mu'], 0), 
    ).assign(year_fe = lambda x: x['year_fe'] + 0.1*(x['year']-x['cohort_year']))

    cross_join['tau_cum'] = cross_join.groupby('unit')['tau'].transform(np.cumsum)
    cross_join = cross_join.assign(dep_var = lambda x: 2010-x['cohort_year'] + 
    x['unit_fe'] + x['year_fe'] + x['tau_cum'] + x['error'])
    cross_join['rel_year'] =  cross_join['year'] - cross_join['cohort_year']

    
    return cross_join

sim_df = make_data(500, 40)
sim_df.head()

	unit	year	state	unit_fe	mu	year_fe	cohort_year	error	treat	tau	tau_cum	dep_var	rel_year
0	0	1980	12	2.770007	1	-1.443538	1998	-0.883541	0	0	0	12.442927	-18
1	0	1981	12	2.770007	1	-3.140326	1998	1.044140	0	0	0	12.673821	-17
2	0	1982	12	2.770007	1	-2.430673	1998	0.276021	0	0	0	12.615355	-16
3	0	1983	12	2.770007	1	-4.253994	1998	0.181049	0	0	0	10.697061	-15
4	0	1984	12	2.770007	1	-3.347429	1998	-2.104055	0	0	0	9.318523	-14

We can now plot the staggered nature of the imagined treatment regime.

Code

fig, ax = plt.subplots(figsize=(10, 6))

for unit in sim_df['unit'].unique()[0:100]:
    temp = sim_df[sim_df['unit'] == unit]
    ax.plot(temp['year'], temp['dep_var'], alpha=0.1, color='grey')

sim_df.groupby(['cohort_year', 'year'])[['dep_var']].mean().reset_index().pivot(index='year', columns='cohort_year', values='dep_var').plot(ax=ax)
ax.axvline(1986)
ax.axvline(1992, color='orange')
ax.axvline(1998, color='green')
ax.axvline(2004, color='red')
ax.set_title("Simulated Cohorts Homogenous Treatment Effects \n All Eventually Treated", fontsize=20)
ax.legend()

<matplotlib.legend.Legend at 0x10ecf1190>

This data will present problems for the vanilla TWFE estimator in part because we can see how each cohort receives a treatment and there are periods in the data when no group is in the “control”. We can see how this plays out with de-meaning TWFE strategy.

fit_twfe = pf.feols(
    "dep_var ~ i(rel_year, ref=-1.0) | state + year",
    sim_df,
    vcov={"CRV1": "state"},
)


figsize = [1200, 400]
fit_twfe.iplot(
    coord_flip=False,
    title="TWFE-Estimator",
    figsize=figsize,
    xintercept=18.5,
    yintercept=0,
).show()

/Users/nathanielforde/mambaforge/envs/pymc_causal/lib/python3.11/site-packages/pyfixest/estimation/feols_.py:1987: UserWarning: 
            The following variables are collinear: ['C(rel_year, contr.treatment(base=-1.0))[T.18]', 'C(rel_year, contr.treatment(base=-1.0))[T.19]', 'C(rel_year, contr.treatment(base=-1.0))[T.20]', 'C(rel_year, contr.treatment(base=-1.0))[T.21]', 'C(rel_year, contr.treatment(base=-1.0))[T.22]', 'C(rel_year, contr.treatment(base=-1.0))[T.23]'].
            The variables are dropped from the model.
            
  warnings.warn(

This is not the expected pattern. For contrast, consider an alternative estimator.

fit_lpdid = lpdid(
    data=sim_df,
    yname="dep_var",
    gname="cohort_year",
    tname="year",
    idname="unit",
    vcov={"CRV1": "state"},
    pre_window=-17,
    post_window=17,
    att=False,
)

fit_lpdid.iplot(
    coord_flip=False,
    title="Local-Projections-Estimator",
    figsize=figsize,
    yintercept=0,
    xintercept=18.5,
).show()

The initial TWFE estimate is frankly bizarre and utterly skewed. Something dreadful has gone wrong under the hood. For contrast, we’ve included the Local Projections estimator from the pyfixest to show that we can recover the actual treatment effect under this event study with alternative strategies. However, there is more machinary involved in the local-projections estimator. Instead we want show how to use mundlak devices to recover more reasonable estimates. No fancy estimators, just more regressions.

Woolridge argues for this approach when he states that

[T]hat there is nothing inherently wrong with TWFE, which is an estimation method. The problem with how TWFE is implemented in DiD settings is that it is applied to a restrictive model… - Two-Way Fixed Effects, the Two-Way Mundlak Regression, and Difference-in-Differences Estimators

We want to explore different ways of isolating the treatment effects under a variety of model specifications that controls for time-constant treatment intensities, covariates, and interactions between them.

Fitting a Variety of Models

sim_df['unit_mean'] = sim_df.groupby('unit')['treat'].transform(np.mean)

sim_df['state_mean'] = sim_df.groupby('state')['treat'].transform(np.mean)

sim_df['cohort_mean'] = sim_df.groupby('cohort_year')['treat'].transform(np.mean)

sim_df['time_mean'] = sim_df.groupby('year')['treat'].transform(np.mean)


model_twfe = bmb.Model(f"""dep_var ~ 1  + time_mean:state_mean + C(rel_year, Treatment(reference=-1))""", sim_df)

idata_twfe = model_twfe.fit( inference_method="nuts_numpyro",
    idata_kwargs={"log_likelihood": True})


model_twfe1 = bmb.Model(f"""dep_var ~ 1  + time_mean* state_mean + C(cohort_year) + C(rel_year, Treatment(reference=-1))""", sim_df)

idata_twfe1 = model_twfe1.fit( inference_method="nuts_numpyro",
    idata_kwargs={"log_likelihood": True})


model_twfe2 = bmb.Model(f"""dep_var ~ 1  + cohort_mean: state_mean + C(rel_year, Treatment(reference=-1))""", sim_df)

idata_twfe2 = model_twfe2.fit( inference_method="nuts_numpyro",
    idata_kwargs={"log_likelihood": True})

model_twfe3 = bmb.Model(f"""dep_var ~ (1| year)  + state_mean + C(rel_year, Treatment(reference=-1))""", sim_df)
idata_twfe3 = model_twfe3.fit( inference_method="nuts_numpyro",
    idata_kwargs={"log_likelihood": True})

These latter models will recover the appropriate treatment effects with slight variations due to the adapted functional form.

Code

fig, axs = plt.subplots(4, 1, figsize=(10, 15), sharey=True)
axs = axs.flatten()
plot_event_study(idata_twfe, ax=axs[0], model='Additive Mundlak')
plot_event_study(idata_twfe1, ax=axs[1], color='red', model='Mundlak State & Time Interactions')
plot_event_study(idata_twfe2, ax=axs[2], color='green', model='Mundlak Cohort & State Interactions')
plot_event_study(idata_twfe3, ax=axs[3], color='purple', model='Hierarchical Time + Mundlak State')
axs[0].set_title(str(model_twfe.formula))
axs[1].set_title(str(model_twfe1.formula))
axs[2].set_title(str(model_twfe2.formula))
axs[3].set_title(str(model_twfe3.formula))
xs = np.linspace(0, 45, 45)
xs_centered = xs - 22
true_effect = np.where(xs_centered <= 0, 0, (xs_centered +1))
axs[0].plot(xs, true_effect, color='k', alpha=0.3, label='True effect')
axs[1].plot(xs, true_effect, color='k', alpha=0.3, label='True effect')
axs[2].plot(xs, true_effect, color='k', alpha=0.3, 
label='True effect')
axs[3].plot(xs, true_effect, color='k', alpha=0.3,
label='True effect')
axs[0].legend()
axs[1].legend()
axs[2].legend()
axs[3].legend()

<matplotlib.legend.Legend at 0x2a8a0eb10>

Note how the naive mundlak approach also exhibits odd behaviour as we saw in the TWFE estimation routine above. Adding additional interactions and controlling for the staggered launch dates seems to help isolate the real pattern in the data. We’ve seen approximate success in a number of these richly parameterised versions of simple event study regressions. But the idiosyncracies of any one sample will contort and distort the estimates away from the true values. We might hope for more stability in expectation over repeated draws from this kind of data-generating process. Consier the following Bootstrapping estimation routine.

Evaluating Robustness of Functional Form

Code

fig, axs = plt.subplots(2, 1, figsize=(10, 15))
axs = axs.flatten()
event_coefs = []
event_coefs1 = []

def fit_ols(formula, df_het):
    twfe_model_ols = smf.ols(formula, data=df_het).fit()
    twfe_model_ols.summary()
    param_est = pd.DataFrame(twfe_model_ols.params, columns=['estimate']).iloc[1:-1]
    param_est['index_number'] = list(range(len(param_est)))
    temp = (param_est.reset_index())
    param_est = temp[(~temp['index'].str.contains(':')) & (temp['index'].str.contains('rel'))]
    param_est.reset_index(inplace=True)
    try:
        param_est['rel_year'] =(param_est['index'].str.split('[', expand=True)[1].str.replace('T.', '')
        .str.replace(']', ''))
    except Exception as e:
        param_est['rel_year'] = param_est['level_0'] - 22
    param_est['rel_year'] = param_est['rel_year'].astype(int)
    param_est.set_index('rel_year', inplace=True)
    return param_est

for i in range(100):
    df_het = make_data(500, 40)
    df_het['state_mean'] = df_het.groupby('state')['treat'].transform(np.mean)
    df_het['time_mean'] = df_het.groupby('year')['treat'].transform(np.mean)
    df_het['cohort_mean'] = df_het.groupby('cohort_year')['treat'].transform(np.mean)
    df_het['unit_mean'] = df_het.groupby('unit')['treat'].transform(np.mean)

    formula = """ dep_var ~ 1 + time_mean + state_mean + cohort_mean  + C(rel_year, Treatment(reference=-1)) """
    formula1 = """ dep_var ~ 1 + C(cohort_year) + time_mean:unit_mean + C(rel_year, Treatment(reference=-1))"""

    param_est = fit_ols(formula, df_het)
    axs[0].plot(param_est['estimate'], color='blue', alpha=0.3)
    event_coefs.append(param_est['estimate'])
    param_est1 = fit_ols(formula1, df_het)
    axs[1].plot(param_est1['estimate'], color='blue', alpha=0.3)
    event_coefs1.append(param_est1['estimate'])

bootstrap_df = pd.DataFrame(event_coefs)
mean_df = bootstrap_df.mean(axis=0)
mean_df.index = param_est.index
mean_df.plot(ax=axs[0], color='red')
axs[0].set_title(f"""Bootstrapped Estimates of Demean Event Study DGP \n {formula}""")

bootstrap_df1 = pd.DataFrame(event_coefs1)
mean_df = bootstrap_df1.mean(axis=0)
mean_df.index = param_est1.index
mean_df.plot(ax=axs[1], color='red')
axs[1].set_title(f"""Bootstrapped Estimates of Event Study DGP \n
{formula1}""");

Here we see the importance of robustness tests in estimation techniques. In the above plot we see something like the correct pattern emerging in expectation, but absurd estimates occuring on individual cases. Whereas the bottom plot appears much more consistent across multiple draws from the data generating process. This points to something about the justification required for a credible result. Asymptotic accuracy is generally not sufficient for inducing credibility in any particular analysis of finite data.

We can also compare these models on standard adequacy measures.

az.compare({'fe_mundlak_naive': idata_twfe, 
            'mundlak_state_time_interactions_cohort': idata_twfe1, 
            'mundlak_cohort_state_interactions': idata_twfe2, 
            'mundlak_state_hierarchical_year': idata_twfe3})

	rank	elpd_loo	p_loo	elpd_diff	weight	se	dse	warning	scale
mundlak_state_hierarchical_year	0	-35912.506751	72.264053	0.000000	0.815835	72.052040	0.000000	False	log
mundlak_state_time_interactions_cohort	1	-36427.688626	53.416108	515.181875	0.184160	74.316511	41.158294	False	log
mundlak_cohort_state_interactions	2	-38253.154180	46.082888	2340.647429	0.000027	78.054836	49.578719	False	log
fe_mundlak_naive	3	-46324.546269	38.489303	10412.039518	0.000000	86.784244	97.445902	False	log

Again, we can plot the predictive performance of these models. Here we compare two via posterior predictive checks. Notable we see how the model with the better predictive performance also correctly estimates the treatement effects in the above plots.

model_twfe3.predict(idata_twfe3, kind='pps')
model_twfe.predict(idata_twfe, kind='pps')
fig, axs = plt.subplots(2, 1, figsize=(10, 12))
axs = axs.flatten()
az.plot_ppc(idata_twfe3, ax=axs[0])
axs[0].set_title("PPC checks for Mundlak State Hierarchical Year")
axs[1].set_title("PPC checks for FE Mundlak Naive")
az.plot_ppc(idata_twfe, ax=axs[1]);

It’s certainly possible that parameter recovery accuracy and predictive performance come apart. But in lieu of knowledge of the true data generating processes, the route to credible models is via their calibration against real data.

Conclusion

“A schema may be transported almost anywhere. The choice of territory for invasion is arbitrary; but the operation within that territory is almost never completely so.” - Nelson Goodman in Languages of Art

We’ve now seen various cases of group-level confounding and strategies to address these issues. We’ve seen how they give rise to non-trivial questions about the effects of policy, and what can be learned in the face of group-level confounds. The case of TWFE’s estimation is especially instructive - here we have an estimation technique applied well in limited circumstances, which ran into trouble when applied in contexts with a diverging data generating process. We all lose our way some of the time, but this misstep was due to a broken map - we’d followed the trail well past the intended territory.

Statistical models do not mirror the world, they project patterns in the data. Each rupture between model and world is a gift that highlights deficits in our understanding - an impropriety of projections. It is therefore a delightful reversal to see, in Woolridge’s proof, that the issue is not with the tooling but the thoughtless application of the tool. The unreasonable effectiveness of regression modelling is surely driven by projections of linearity on the world, but it remains compelling in approximation when used well. There are architectural constraints in causal inference - hard unyielding facts, that shape the contours of our design, and structure the steps we can make. The degree to which we can flexibly encode those structures in our models, the more compelling our models will be. It is this recognition of dual fit requirements between world and model - the focus on the conditions for identifiability - that lends causal inference any kind of credibility. The expressive capacity of generalised linear modelling enables us to articulate too many candidate pictures of the world. It’s the business of science to cull those pictures, to find only the plausible representations of reality.

Citation

BibTeX citation:

@online{forde2024,
  author = {Nathaniel Forde},
  title = {Freedom, {Hierarchies} and {Saturated} {Regression} {(WIP)}},
  date = {2024-07-01},
  langid = {en}
}

For attribution, please cite this work as:

Nathaniel Forde. 2024. “Freedom, Hierarchies and Saturated Regression (WIP).” July 1, 2024.